Question: Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{6p^2 - 6}{-10p^2 - 30p + 40}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {6(p^2 - 1)} {-10(p^2 + 3p - 4)} $ $ n = -\dfrac{6}{10} \cdot \dfrac{p^2 - 1}{p^2 + 3p - 4} $ Simplify: $ n = - \dfrac{3}{5} \cdot \dfrac{p^2 - 1}{p^2 + 3p - 4}$ Next factor the numerator and denominator. $ n = - \dfrac{3}{5} \cdot \dfrac{(p - 1)(p + 1)}{(p - 1)(p + 4)}$ Assuming $p \neq 1$ , we can cancel the $p - 1$ $ n = - \dfrac{3}{5} \cdot \dfrac{p + 1}{p + 4}$ Therefore: $ n = \dfrac{ -3(p + 1)}{ 5(p + 4)}$, $p \neq 1$